How do I go about solving for the exact value of this experssion

Question

sec(sin^-1 12/13)

I assume I solve for the arcsin for 12/13 first and the find the secant of what I got, but that doesn't sound right.

Answer 1

You do not actually have to find the angle that has a sine of 12/13, although you could. Think of the reference triangle in the 1st quadrant where the hypotenuse is 13 and the leg opposite the reference angle is 12. That angle will indeed have a sine of 12/13. You can use the Pythagorean Theorem to find he other leg of the triangle (or if you remember some Pythagorean Triples, you will label the missing leg easily as 5, i.e. a 5-12-13 Pythagorean Triple).

Now the secant of that angle is the reciprocal of the cosine. Since the cosine is 5/13, the secant is 13/5.

Here is a Desmos graph showing all of this (the angle is in degrees):

desmos.com/calculator/kk3gwr9lqx